Problem: Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{2}{3}\pi) + i \sin(\frac{2}{3}\pi)}) ^ {20} $
Solution: Let's express our complex number in Euler form first. $ {\cos(\frac{2}{3}\pi) + i \sin(\frac{2}{3}\pi)} = { e^{2\pi i / 3}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{2\pi i / 3}}) ^ {20} = e ^ {20 \cdot (2\pi i / 3)} $ The angle of the result is $20 \cdot \frac{2}{3}\pi$ , which is $\frac{40}{3}\pi$ $\frac{40}{3}\pi$ is more than $2 \pi$ . It is a common practice to keep complex number angles between $0$ and $2 \pi$ , because $e^{2 \pi i} = (e^{\pi i}) ^ 2 = (-1) ^ 2 = 1$ . We will now subtract the nearest multiple of $2 \pi$ from the angle. $ \frac{40}{3}\pi - 12\pi = \frac{4}{3}\pi $ Our result is $ e^{4\pi i / 3}$. Converting this back from Euler form, we get $\cos(\frac{4}{3}\pi) + i \sin(\frac{4}{3}\pi)$.